Theorem subgex 13956

Description: The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.)

Assertion

Ref	Expression
subgex	|- ( G e. Grp -> (SubGrp ` G ) e. _V )

Proof of Theorem subgex

Dummy variables s w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-subg 13950	. . 3 |- SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
2		fveq2 5690	. . . . 5 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
3	2	pweqd 3690	. . . 4 |- ( w = G -> ~P ( Base ` w ) = ~P ( Base ` G ) )
4		oveq1 6082	. . . . 5 |- ( w = G -> ( w ↾s s ) = ( G ↾s s ) )
5	4	eleq1d 2307	. . . 4 |- ( w = G -> ( ( w ↾s s ) e. Grp <-> ( G ↾s s ) e. Grp ) )
6	3, 5	rabeqbidv 2816	. . 3 |- ( w = G -> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } = { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } )
7		id 19	. . 3 |- ( G e. Grp -> G e. Grp )
8		basfn 13389	. . . . . 6 |- Base Fn _V
9		elex 2833	. . . . . 6 |- ( G e. Grp -> G e. _V )
10		funfvex 5707	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
11	10	funfni 5478	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
12	8, 9, 11	sylancr 418	. . . . 5 |- ( G e. Grp -> ( Base ` G ) e. _V )
13	12	pwexd 4313	. . . 4 |- ( G e. Grp -> ~P ( Base ` G ) e. _V )
14		rabexg 4274	. . . 4 |- ( ~P ( Base ` G ) e. _V -> { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } e. _V )
15	13, 14	syl 14	. . 3 |- ( G e. Grp -> { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } e. _V )
16	1, 6, 7, 15	fvmptd3 5793	. 2 |- ( G e. Grp -> (SubGrp ` G ) = { s e. ~P ( Base ` G ) | ( G ↾s s ) e. Grp } )
17	16, 15	eqeltrd 2315	1 |- ( G e. Grp -> (SubGrp ` G ) e. _V )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   ~Pcpw 3685   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   Basecbs 13330   ↾_s cress 13331   Grpcgrp 13782  SubGrpcsubg 13947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-subg 13950

This theorem is referenced by:  isnsg  13982